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Archive for the ‘mathematics’ Category

Two wrongs make a right or wrong — in theory?

Sunday, November 15th, 2015

[ by Charles Cameron — on the (Pythagorean) arithmetic of morals ]

Extermination_of_Evil_Sendan_Kendatsuba 600

Sendan Kendatsuba, one of the guardians of Buddhist law, banishing evil, Tokyo National Museum


What’s right is generally supposed to be positive, while what’s wrong is seen as negative — and as they saying goes, two wrongs don’t make a right.

In effect, that’s saying two negatives don’t make a positive. And if you add them, that’s correct.

But if you multiply two negatives, you get a positive — hunh?

So two wrongs can indeed make a right — that’s the mathematics of vengeance — multiplicative:

And thine eye shall not pity; but life shall go for life, eye for eye, tooth for tooth, hand for hand, foot for foot.

— Deuteronomy 19.13

And it is also true that two wrongs don’t make a right — that’s a mathematics that denies vengeance — additive.

And then there’s the mathematics of forgiveness :

Be not overcome of evil, but overcome evil with good.

— Romans 12.21

Patient men, desirous of the Face of their Lord, who perform the prayer, and expend of that We have provided them, secretly and in public, and who avert evil with good — theirs shall be the Ultimate Abode

— Qur’an 13.22

And what’s most interesting to me in all this, is that the mathematical formulations, additive and multiplicative alike, don’t make a feature of time — where as their moral equivalents tend to introduce time into the equation / situation — in each case, it’s the response to evil, real or potential, that is considered.

It is more important to have equations in your beauty..

Tuesday, November 10th, 2015

[ by Charles Cameron — musings on PAM Dirac and Hedy Lamarr ]

PAM Dirac once famously said, “it is more important to have beauty in one’s equations that to have them fit experiment.” I was reminded of that quotation while musing over this DoubleQuote in the Wild celebrating the birthday yesterday of Hedy Lamarr — stunningly symmetrical movie star and frequency-hopping inventor:

Hedy Lamarr DQ Wild

To the left, the beauty herself, and to the right, the equations in the beauty — if one may take the liberty of saying that thoughts are “in” their thinker’s body — shown here in the form of the patent application Lamarr and composer George Antheil submitted fora “a system for the radio control of airborne torpedoes”.

I put that last phrase in quotes because that’s the way a New York Times piece described their joint invention. Me, I haven’t the foggiest about their inner workings, whereas the outer face of Hedy Lamarr I can see with utmost clarity.

The Astounding Case of the Unequal Equals

Tuesday, November 10th, 2015

[ by Charles Cameron — how two ideas beget a third ]

I’m a Brit, so the motto Mind the Gap is familiar to me — it warns people not to step between platform and train when entering or exiting the Tube

underground_2504657b mind the gap

— as is the London Underground symbol — and for that matter, the glyph for female, earlier the alchemical symbol of Venus.


Today’s Telegraph contains a graphic that interests me a whole lot, under the title Equal Pay Day: 14 ways to visualise the gender pay gap:

telegraph mind the gap


Does it interest me because I’m a fan of the London Underground? Not exactly. The Underground’s graphics, then? I’ll admit to an admiration for the Underground’s justifiably celebrated map


— which I’ve long thought would make a superb extended HipBone Game board, shown above in its 1968 incarnation. But that’s not it either. And although I don’t think gender should influence the pay a person receives for a given quantity or quality of work — Quant and Qualit again, one of the weirdst paradoxes my min d has ever encountered — it’s not the politics of the sign that interests and delights me.


It’s the way two disparate elements — tube and gender graphics — are combined to create a remarkably powerful third:

SPEC DQ mind the gap

It’s a simple, elegant illustration of the intersection of two sets of ideas that Arthur Koestler talked about and that I’m constantly drawing on in my own work:

And yes, I’m aware that technically the equation I’m pointing to happened in two stages, with the substitution of the phrase “mind the gap” for the word “underground” in the blue bar of the tube logo — but the bar was also used, among other things, for the names of individual stations, so I don’t consider that much of a creative leap.

No, it’s the juxtaposition of underground logo with the gender sign by means of a conflation of their respective circles that’s impressive, coupled with the pun on the gender-specific and underground meanings of “mind the gap”.

That’s the very essence of the intersections Koestler was talking about, and illustrates as vividly as I know how, the power released by such an intersection.


Oh, and yes — fair play clearly requires fair pay.

And it’s Hedy Lamarr‘s birthday, dammit.

Sunday surprise: mathematical religion

Sunday, November 8th, 2015

[ by Charles Cameron — divine geometry in Japan ]

The sacred nature of number should not surprise us, though it often does. Here is the Neoplatonist Proclus, as quoted in Dantzig, Number: the Language of Science, p.78:

It is told that those who first brought out the irrationals from concealment into the open perished in shipwreck, to a man. For the unutterable and the formless must needs be concealed. And those who uncovered and touched this image of life were instantly destroyed and shall remain forever exposed to the play of the eternal waves.


Here are two offerings from a Buddhist shrine in Japan:

Japanese Buddhist Offering 19.50


Japanese Buddhist Offering 19.47


I found them in this delightful account of the Soroban or Japanese abacus:

The section on mathematical offerings begins around the 19.40 mark and goes to 20.18 — but the whole video is worth your attention.

It is always good to find oneself in good company

Friday, October 30th, 2015

[ by Charles Cameron — Aristotle, the Ismailis, CG Jung ]

Two is the first number:

From the tenth-century Rasa’il Ikhwan al-Safa’ (Epistles or Treatises of the Brethren of Purity) as found in Seyyed Hossein Nasr & Mehdi Aminrazavi, eds., An Anthology of Philosophy in Persia, vol. 2, Ismaili Thought in the Classical Age:

We say that one is the source and generator of the numbers because when one is removed from existence all the numbers are removed with it, but when the numbers are removed from existence, one is not removed. We say that two is the first whole number because numbers are a plurality of ones, and the first plurality is two.


I am delighted to find myself in the company of the Ismailis, having written in On two, one, seven plus or minus, and ten – towards infinity:

When I worked as senior analyst in a tiny think-shop, my boss would often ask me for an early indicator of some trend. My brain couldn’t handle that — I always needed two data points to see a pattern, and so I coined the mantra for myself, two is the first number.

One isn’t a number until there are two, because it’s limitless across all spectra and unique, and because it is its own, only context.

One isn’t a number unless there’s a mind to think of it — in which case it’s already an abstraction within that mind, and thus there are, minimally, two. At which point we are in the numbers game, and there may be many, many more than two — twenty, or plenty, or plenty-three, or the cube root of aleph null, or (ridiculous, I know) infinity-six…

Go, figure.

Two is the first number, because the two can mingle or separate, duel or duet: either way, there’s a connection, a link between them.

And in Of a non-comparative use of the DoubleQuotes method:

Putting it bluntly, one point is pointless — things could go anywhere from there. Two points suggest a line, a link, a connection — a possible, maybe even plausible, trend.

It’s that sense I have of two being the beginning of thought that makes me so fond of the DoubleQuotes format — and of Arthur Koestler‘s insight about creation occurring at the intersection of two spheres..


If I’m a fundamentalist about anything, it’s the notion that it takes two to tango!



From CG Jung, A Psychological Approach to the Dogma of the Trinity, in Psychology and Western Religion:

The number one claims an exceptional position, which we meet again in the natural philosophy of the Middle Ages. According to this, one is not a number at all; the first number is two. Two is the first number because, with it, separation and multiplication begin, which alone make counting possible. With the appearance of the number two, another appears alongside the one, a happening which is so striking that in many languages “the other” and “the second” are expressed by the same word.

And from Leonardo Tarán, Speusippus of Athens: A Critical Study With a Collection of the Related Texts and Commentary:

Now we know that Speusippus did consider one to be the first odd number, whereas Aristotle thinks that two is the first number and explicitly denies that one is a number. [ … ]

Concerning the One, then, the differences between Speusippus and Aristotle amount to this: For the latter the One is not a number, nor a separately existing entity, but the principle of number because, each number being a plurality of abstract monads, the One is its measure.


As Emerson wrote:

The world is young; the former great men call to us affectionately.

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